Supercooled liquids approaching the glass transition show increasingly slow dynamics, until eventually they cannot equilibrate in laboratory timescales19. One consequence of this fact is physical ageing, that is, the breakdown of time translation invariance: the correlation C(t,tw) between spontaneous fluctuations of an observable at times t (the final time) and tw (the waiting time) is a non-trivial function of t and tw, as opposed to being a function of the time difference ttw. In many cases, the two-time correlation C(t,tw) in an ageing system separates into a fast, time-translation-invariant contribution Cfast(ttw) and a slow contribution Cslow(t,tw) (ref. 20): C(t,tw)=Cfast(ttw)+Cslow(t,tw). For some systems, the slow part of the correlation has the form20 Cslow(t,tw)=Cslow(h(t)/h(tw)), where h(t) is some monotonically increasing function. For example, in the case of domain growth, h(t) is proportional to the domain size20. In what follows, only the slow part of the correlation is considered, and any effects due to the fast part of the dynamics are ignored.

Recently, it has been proven that, in the limit of long times, the dynamics of a class of spin-glass models is invariant under global reparametrizations th(t) of the time15. This result has been used to predict the existence of a Goldstone mode in the non-equilibrium dynamics, associated with smoothly varying local fluctuations in the reparametrization of the time (refs 16,17). These fluctuations have been physically interpreted to represent local fluctuations of the age of the sample16,17. In the cases where the global two-time correlation shows h(t)/h(tw) scaling, a simple Landau-theory approximation for the dynamical action predicts16,17,18 that the full probability distribution ρ(Cr(t,tw)) of local correlations Cr(t,tw) depends on the times t,tw only through the values of the global correlation Cglobal(t,tw). However, this scaling of ρ(Cr(t,tw)) with Cglobal(t,tw) was also found in the coarsening dynamics of the O(N) ferromagnet, where the time reparametrization symmetry is not present21. It is not known whether off-lattice, quasi-realistic models, describing structural glasses, show the same time reparametrization symmetry as spin glasses, or whether their dynamics shows any evidence of the Goldstone mode associated with this symmetry. A first test for this statement would be to check whether ρ(Cr(t,tw)) scales with Cglobal(t,tw). If this test fails, then the presence of the time reparametrization symmetry can be excluded.

Simulations of fluctuations in glass-forming liquids have mostly focused on the (equilibrium) supercooled liquid, and on determining the spatial correlation of fluctuations between different points in space12,13,14. In ref. 22 the ageing regime was studied, but only the spatial correlations of fluctuations were measured, and in ref. 18 spin-glass and kinetically constrained lattice models were studied.

Here, we present the first detailed characterization of the statistics of local fluctuations in the ageing of a continuous-space, quasi-realistic structural glass model. Numerically simulating the (non-equilibrium) ageing regime allows us to address many experiments working in this regime that probe dynamical heterogeneities microscopically7,8,9,10,11. We focus here on determining the statistical distribution of fluctuations at one point in space, for various reasons: (1) to make direct contact with experiments using local probes to study dynamical heterogeneities, which also obtain this kind of distribution5,6,10; (2) to obtain additional physical information beyond the second moment of the fluctuations and (3) to test whether the probability distribution of local fluctuations in the ageing regime depends on times only through the value of Cglobal(t,tw).

We probe individual particle displacements along one direction Δxj(t,tw)=xj(t)−xj(tw) (where j is the particle index), and also local, coarse-grained two-time functions: the correlator

and the mean square displacement

Here we consider a coarse-graining cubic-shaped box Br of side l around the point r in the system, and the sums run over the N(Br) particles present at the waiting time tw in Br. We choose a value of q that corresponds to the main peak in the structure factor S(q) of the system, q=7.2 in Lennard-Jones (LJ) units.

These definitions are inspired by the analogous definitions in the case of spin glasses16,17, and can be applied to analyse data obtained both from simulations and from confocal microscopy experiments. The global quantities Cglobal(t,tw) (incoherent part of the intermediate scattering function) and Δglobal(t,tw) (mean square displacement) are defined by extending the sum to the whole system in equations (1) and (2) respectively.

We carried out 250 independent molecular-dynamics runs for the binary LJ system of ref. 23, which has a mode-coupling critical temperature Tc=0.435. A system of 8,000 particles was equilibrated at a temperature T0=5.0, then instantly quenched to T=0.4, and finally it was allowed to evolve for 105 LJ time units. The origin of times was taken at the instant of the quench.

In Fig. 1a,b we present our results for the probability distribution ρ(Cr(t,tw)) of the local intermediate scattering function for waiting times tw=30.20,…,30,200, and final times t chosen so that Cglobal(t,tw){0.1,0.3,0.5,0.7}. We observe that the data approximately collapse for each value of Cglobal(t,tw) (a less clear collapse is observed at constant Δglobal(t,tw); details of this comparison will be presented elsewhere). This collapse at constant Cglobal(t,tw) is also observed in simulations in a three-dimensional spin-glass model, but in the case of the spin-glass model the collapse is more precise than here. Unlike the case of the three-dimensional spin-glass model, the position of the peak in the distribution ρ(Cr) is strongly dependent on the value of Cglobal(t,tw). The distribution ρ(Cr(t,tw)) evolves gradually from being highly skewed and non-gaussian for Cglobal(t,tw)=0.7 to being unskewed and very close to gaussian for Cglobal(t,tw)=0.1. Notice here that the distributions of local observables are also expected to become more gaussian as Cglobal(t,tw) is increased beyond Cglobal(t,tw)≈0.7, that is, in the quasi-equilibrium regime corresponding to the first step in the two-step relaxation. This is indeed observed in experiments probing fluctuations in dipole moments of nanometre-scale regions11 and also in our simulations, in the probability distributions ρx) of one-dimensional displacements.

Figure 1: Probability distributions.
figure 1

a,b, ρ(Cr(t,tw)) measured for 30.20≤tw≤30,200, plotted for final times t chosen so that Cglobal(t,tw){0.1,0.3,0.5,0.7}. Coarse graining size l≈0.11L (with L≡ linear size of the simulation box). The curves collapse into four groups, corresponding to Cglobal=0.7,0.5,0.3,0.1 (ordered from highest to lowest value of Cr at the peak). A gaussian fit to the data for C=0.1 is also shown. Linear scale (a). Logarithmic scale (b). c, ρx(t,tw)) for 30.20≤tw≤30,200, with t chosen so that Cglobal(t,tw){0.1,0.3,0.5,0.7}. The curves collapse into four groups, corresponding to Cglobal=0.1,0.3,0.5,0.7. d, Tails of ρx(t,tw)), for C=0.5 and tw=30.20,302,3,020,30,200 (from narrower to wider tail). Symbols: results from simulation. Lines: fits to the data for |Δx|>0.5 using ρx(t,tw))≈Nexp(−|Δx/a|β).

To characterize the weak dependence of the probability distributions on waiting time at fixed Cglobal(t,tw), in Fig. 2a we plot the centred second moment of the distributions ρ(Cr) as a function of waiting time, for fixed Cglobal(t,tw){0.1,0.3,0.5,0.7}. The dependence on tw is so weak that both a logarithmic form and a power-law form (with powers in the range 0.01–0.07) provide a good fit. We can explain the fact that ρ(Cr) does show some dependence on tw for fixed Cglobal by the presence of a time-dependent dynamic correlation length. As in the case of simulations of spin glasses17, the dynamic correlation length in the present system grows very slowly as a function of tw, but for the timescales of the simulation it is not yet larger than the size of the coarse-graining box used24. Thus, some of the fluctuations are averaged out, and the width of the distribution is reduced. This effect is stronger for shorter tw, consistent with the trend shown in Fig. 2a.

Figure 2: Dependence of probability distributions on the waiting time.
figure 2

Evolution of the probability distributions, as a function of tw, at constant Cglobal(t,tw){0.1,0.3,0.5,0.7}. a, Second moment of ρ(Cr(t,tw)), together with fits to the functional forms: m0(tw)a (full lines) and m0log(tw/t0) (dotted lines). b, Stretching exponent β for the tails of ρx), as a function of the waiting time tw, at constant Cglobal(t,tw){0.1,0.3,0.5,0.7} (the lines are guides to the eye). The error bars indicate the statistical errors in β.

In Fig. 1c,d, we present our results for the probability distribution ρx(t,tw)) of the particle displacements Δxj(t,tw)=xj(t)−xj(tw) along one direction. In Fig. 1c, we can observe that these data also approximately collapse for each value of Cglobal(t,tw). In Fig. 1d, we have a closer look at the tails of ρx(t,tw)). We find that the distribution is non-gaussian, as was observed in experiments in colloidal glasses in the supercooled regime5,6. We can fit the tails of the distribution with a nonlinear exponential form ρx)≈Nexp(−|Δx/a|β), and they become more prominent as tw grows (for constant Cglobal(t,tw)). Indeed, as shown in Fig. 2b, the exponent β decreases from β>1 (‘compressed exponential’) at short tw to β≈0.8 (‘stretched exponential’) at much longer tw.

To summarize, we have presented the first detailed characterization of the probability distributions of non-equilibrium fluctuations in the ageing regime in a continuous-space, quasi-realistic structural glass model. Our main result is that the probability distributions for the local fluctuating two-time quantities are, to a first approximation, invariant when the global intermediate scattering function Cglobal(t,tw) is kept constant. This behaviour is similar to the behaviour found in the non-equilibrium dynamics of short-range spin-glass models16,17 and some kinetically constrained lattice models18, and in the coarsening dynamics of the O(N) model21. As a consequence, our results cannot rule out the presence of a Goldstone mode associated with local fluctuations in the age of the sample, but alternative interpretations are still possible21. Besides this simple scaling, our results provide detailed predictions for the statistical properties of fluctuations in ageing structural glasses. These predictions can be directly tested by applying a similar analysis to experimental data from confocal microscopy in colloidal glass systems5,6,7, and also possibly by analysing atomic force microscopy experiments probing nanoscale polarization fluctuations8,9,10,11.